Hausdorff dimension of the Apollonian gasket

TL;DR

This paper introduces an efficient method to compute the Hausdorff dimension of the Apollonian gasket with unprecedented precision, providing rigorous error bounds and potential applicability to other fractals.

Contribution

The authors develop a novel computational approach that accurately estimates the Hausdorff dimension of the gasket to 128 decimal places with rigorous error bounds.

Findings

Dimension computed to 128 decimal places

Method provides rigorous error bounds

Approach generalizes to other parabolic fractals

Abstract

The Apollonian gasket is a well-studied circle packing. Important properties of the packing, including the distribution of the circle radii, are governed by its Hausdorff dimension. No closed form is currently known for the Hausdorff dimension, and its computation is a special case of a more general and hard problem: effective, rigorous estimates of dimension of a parabolic limit set. In this paper we develop an efficient method for solving this problem which allows us to compute the dimension of the gasket to 128 decimal places and rigorously justify the error bounds. We expect our approach to generalise easily to other parabolic fractals.